What Is the Value of Constant k in This Probability Distribution Function?

[SMC]

Homework Statement

The technical specification of a particular electrical product states that the probability of its failure with time is given by the function:

f(t) = 1 - ke^(-t/t₀) if 0 < t < t_max
f(t) = 0 if t > t_max

where t is the time of service in years, and the constant t₀ = 100 years defines the characteristic deterioration time. If the maximal life span of the product t_max is 10 years,
find the value of constant k , explaining its meaning,

Homework Equations

i thought i was supposed to find the integral of 1 - ke^(-t/t₀) with respect to t between 0 and 10 years and put the answer equal to 1. then find k from that. [/B]

The Attempt at a Solution

now this is a past exam paper question so i actually have the partial solution to the question which says that:

k = -9/(t₀(e^(-10/t₀) - 1))

but what i get is k = -9/(t₀e^(-10/t₀))

I'm pretty sure I'm doing the integral correctly so there must be some step I'm missing that accounts for that extra -1 in the denominator.
[/B]

 

[mfb]

Did the past exam question have exactly the same problem statement?
The second solution looks odd. It gives wrong results for very large t₀, for example. How did you get it?

Why do both solutions depend on t? They should not do that. Is that t_max?

 

[SMC]

yeah sorry that t should be t_max = 10... I've edited it

 

[SMC]

i just did the integral 0 and 10 of the function given and put it equal to 1 because the probability of it failing within a 10 year period is 100%. once I've done the integral I get:

10 - k*e^(-10/t₀)*(-t₀) = 1 (int. between 0 and 10 of 1 is 10 and int. between 0 and 10 of k*e^(-t/t₀) is k*e^(-10/t₀)*(-t₀) )

then i find k from that. is this wrong?

 

[SMC]

OMG I just realized what I did! I'm really sorry! never mind! please ignore!

 

[SMC]

i considered e^0 = 0 I've made this mistake quite a few times especially in integrlas!